leakage at nodes, (3) total pump heads.
The flow change penalty is introduced to all iteration steps to prevent solution oscillation, which occurs between two similar solutions in the final iteration steps and prevents convergence. It was found out that the flow change penalty helps to reach the optimal solution in less iteration steps. Several scenarios (cases) are analysed, constraints are increasingly
implemented into scenarios.
Test networks: (1) Complex WDS with 3 pressure zones (incl. 15 nodes). 94. Ghaddar et al. (2014)
SO
Optimal pump operation using Lagrangian decomposition with improved limited discrepancy search (ILDS) algorithm.
Objective (1): Minimise (a) the pump operating costs (energy consumption charge).
Constraints: (1) Upper bound for pipe flows, (2) pump must be on for the water to flow in the corresponding pipe, (3) min/max tank water levels, (4) nonnegativity for pipe flows, (5) min length of time for a pump to be on, (6) min length of time for a pump to be off, (7) max number of pump switches, (8) no deficit in tanks at the end of the simulation period.
Decision variables: (1) Pipe flows, (2) pipe headlosses, (3) node pressures, (4) pump statuses (binary, 0 = pump off, 1 = pump on).
Water quality: N/A. Network analysis: EPANET (EPS). Optimisation method: Lagrangian
decomposition combined with ILDS.
Lagrangian decomposition, which is a relaxation, breaks the original problem into smaller subproblems. Due to the relaxation of the original problem, the solutions of the subproblems may not be feasible for the original problem. Hence, a heuristic ILDS is used to find feasible solutions. The ILDS provides an upper bound on the optimal objective function value, while the Lagrangian relaxation provides a lower bound, so the proposed approach provides solutions of guaranteed quality.
The approach is compared with the MILP relaxation of the original MINLP problem, which is solved by CPLEX.
Time horizon is 24 hours, and the decisions to turn a pump on or off are made at 30 minute intervals.
Two electricity pricing schemes are used. First, a fixed day/night scheme; second, a dynamic scheme with prices changing every 30 minutes. The results show that the ILDS can find better solutions than CPLEX in
significantly less time. Optimised pump schedules typically lead to a decrease in tank water levels.
An impact of electricity pricing schemes on the pump operating costs is evaluated. The dynamic pricing results in up to 34% of cost reduction. Test networks: (1) Small network with 1 reservoir, 2 pumps, 2 tanks (incl. 1
node), (2) Poormond network (incl. 47 nodes) adapted from Richmond network (Giacomello et al. 2013).
95. Goryashko and Nemirovski (2014) SO
Optimal pump operation with demand uncertainty using LP.
Objective (1): Minimise (a) the pump operating costs (including two
components: energy consumption charge and the price of water).
Constraints: (1) Bounds on tank levels, (2) bound on pump capacity, (3) bound on source capacity.
Decision variables: (1) The amount of water pumped into the system during a time interval.
Water quality: N/A. Network analysis: Explicit mathematical formulation/ EPANET (EPS). Optimisation method: MOSEK software (MOSEK 2014) using LP.
The original problem of minimisation of pumping cost is simplified to a LP problem, in which the demands are treated as uncertain. To cater for demand uncertainty, the robust counterpart methodology is employed, which involves obtaining the ‘worst-case’ cost over all possible data from the ‘uncertainty set’, ensuring that all the constraints are satisfied for all realisations of the demands. Using the robust counterpart methodology, the uncertain LP model is converted to a linearly adjustable robust counterpart. The results obtained are referred to as linear robust optimal (LRO) policy.
Time horizon is 24 hours divided into 1-hour intervals.
The obtained LRO policy with the uncertainty level set to 20% is tested in EPANET to ensure the appropriate hydraulic behaviour. For testing purposes, the demands were perturbed in EPANET. The results show that the warnings
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in EPANET (negative pressure etc.) start appearing when the perturbations become as large as 50%.
Test networks: (1) Anytown network (incl. 19 nodes) (Walski et al. 1987) with modifications.
96. Ibarra and Arnal (2014) SO
Optimal pump operation using parallel programming techniques and MIP.
Objective (1): Minimise (a) the pump operating costs (energy consumption charge).
Constraints: (1) Min/max operational tank volumes, (2) the number of start/stop events of the pumps.
Decision variables: (1) Pump statuses (binary, 0 = pump off, 1 = pump on during a time interval), (2) special binary
variables Ai and Pi to model start/stop
events of the pumps (they are used to reduce the number of start/stop events).
Water quality: N/A. Network analysis: Explicit mathematical formulation, simplified hydraulic equations (unsteady state). Optimisation method: COIN-OR libraries (COIN-OR 2014) using branch and bound method and demand prediction.
The optimisation problem is formulated as a MIP problem. Time horizon is 24 hours.
The near real-time optimal pump scheduling is proposed based on the demand forecast. The demand forecast is determined every hour for the next 24 hours and the next 7 days using the seasonal autoregressive integrated moving average (SARIMA) (Makridakis et al. 2008) models from the statistical time series theory.
The parallel programming is implemented on both shared and distributed memory multiprocessors. The stochastic scenario tree evaluation and multisite problems (multiple networks controlled from a single control centre) are solved.
Test networks: (1) WDS of Granada, Spain. 97. Hashemi et al. (2014)
SO
Optimal pump operation considering variable speed pumps using ACO.
Objective (1): Minimise (a) the pump operating costs (energy consumption charge).
Constraints: (1) Volume deficit in tanks at the end of the simulation period.
Decision variables: (1) Pump speeds for each interval.
Water quality: N/A. Network analysis: EPANET (EPS). Optimisation method: Ant system iteration best (ASib) algorithm.
Time horizon is 24 hours divided into 1-hour intervals.
Sensitivity analysis to find the best performing values of ASib stochastic
parameters is performed.
For the Richmond network, the results with single speed pumps are compared to the results with variable speed pumps. Cost savings of about 10% are obtained for the network with variable speed pumps.
For the Anytown network, the size of the search space is reduced using two approaches, ‘Replacing reservoir’ (RR) and ‘In-station scheduling’ (ISS). RR involves replacing one of the pumping stations by the reservoir and
optimising head and flow supplied by that reservoir. The decision variable is the water level. ISS involves transforming obtained heads and flows to a pump schedule. The search space is reduced more than 1038 times.
Test networks: (1) Simplified Richmond WDS (incl. 13 nodes) (Van Zyl et al. 2004), (2) optimised design of the Anytown network (incl. 22 nodes) (Murphy et al. 1994).
98. Kurek and Ostfeld (2014) MO
Optimal operation of drinking WDSs including pumping cost and water quality objectives using SPEA2.
Objective (1): Minimise (a) the pump operating costs (energy consumption charge).
Objective (2): Minimise (a) the evaluation function of disinfectant concentrations at monitoring nodes.
Constraints: (1) Pressure at nodes, (2) tank volume surplus/deficit at the end of the
Water quality: Disinfectant (i.e. chlorine). Network analysis: EPANET (EPS). Optimisation method: SPEA2 (Zitzler et al. 2001).
Variable speed pumps are considered.
Time horizon is 72 hours divided into 1-hour intervals. Only the last 24 hours are used to evaluate the values of objective functions and constraints in order to minimise the effect of initial conditions.
Tradeoffs between energy consumed by pumps and water quality are obtained: more energy consumed by pumps results in better water quality, conversely, limiting the amount of energy consumed by pumps results in deterioration of water quality.